Solution of Post-Buckling & Limit Load Problems, Without Inverting the Tangent Stiffness Matrix & Without Using Arc-Length Methods

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1 Copyright 214 Tech Science Press CMES, vol.98, no.6, pp , 214 Solution of Post-Buckling & Limit Load Problems, Without Inverting the Tangent Stiffness Matrix & Without Using Arc-Length Methods T.A. Elgohary 1, L. Dong 2, J.L. Junkins 3 and S.N. Atluri 4 Abstract: In this study, the Scalar Homotopy Methods are applied to the solution of post-buckling and limit load problems of solids and structures, as exemplified by simple plane elastic frames, considering only geometrical nonlinearities. Explicitly derived tangent stiffness matrices and nodal forces of large-deformation planar beam elements, with two translational and one rotational degrees of freedom at each node, are adopted following the work of [Kondoh and Atluri 1986)]. By using the Scalar Homotopy Methods, the displacements of the equilibrium state are iteratively solved for, without inverting the Jacobian tangent stiffness) matrix. It is well-known that, the simple Newton s method and the Newton-Raphson iteration method that is widely used in nonlinear structural mechanics), which necessitates the inversion of the Jacobian matrix, fails to pass the limit load as the Jacobian matrix becomes singular. Although the so called arc-length method can resolve this problem by limiting both the incremental displacements and forces, it is quite complex for implementation. Moreover, inverting the Jacobian matrix generally consumes the majority of the computational burden especially for large-scale problems. On the contrary, by using the presently developed Scalar Homotopy Methods, convergence near limit loads, and in the post-buckling region, can be easily achieved, without inverting the tangent stiffness matrix and without using complex arc-length methods. The present paper thus opens a promising path for conducting post-buckling and limit-load analyses of nonlinear structures. While the simple Williams toggle is considered as an illustrative example in this paper, extension 1 Department of Aerospace Engineering, Texas A&M University, College Station, TX. Student Fellow, Texas A&M Institute for Advanced Study. 2 Corresponding author, Department of Engineering Mechanics, Hohai University, China, dong.leiting@gmail.com 3 Department of Aerospace Engineering, Texas A&M University, College Station, TX. Founding Director, Texas A&M Institute for Advanced Study 4 Center for Aerospace Research & Education, University of California, Irvine. Fellow & Eminent Scholar, Texas A&M Institute for Advanced Study, Texas A&M University, College Station, TX.

2 544 Copyright 214 Tech Science Press CMES, vol.98, no.6, pp , 214 to general finite element analyses of space frames, plates, shells and elastic-plastic solids will be considered in forthcoming studies. Keywords: Nonlinear Algebraic Equations, Post-Buckling, Tangent Stiffness, Newton s Method, Scalar Homotopy Method 1 Solving NAEs without Inverting the Jacobian Matrix A large number of problems in engineering and applied sciences, such as large deformation solid mechanics, fluid dynamics, post-buckling of structural frames, plates, and shells, etc, as characterized by nonlinear differential equations, will lead to a system of nonlinear algebraic equations NAEs) after discretization: F i x 1,...,x n ) =, i = 1,...,n, or in their vector-form: Fx) =, 1) where Fx) R n is a given vector function, and x is the solution vector. To find the unknown vector x R n, the famous Newton s method is widely used to iteratively solve Eq. 1: x k+1 = x k B 1 F k x = a 2) where, a represents the initial guess of the solution, B is the Jacobian tangent stiffness) matrix given by B = F x, and k denotes the number of iteration. In computational solid mechanics, the trend over the past 3 4 years has been to directly derive the tangent stiffness matrix, B, rather than forming the nonlinear equations, Fx) = ) through incremental finite element methods [Crisfield 1983); Kondoh and Atluri 1986); Riks 1972); Wempner 1971)]. Recently, however, [Dai, Yue, and Atluri 214)] directly derived the system of equations, Fx) =, for a von Kármán plate theory using the Galerkin method. As motivated by Eq. 2, [Hirsch and Smale 1979)] also introduced the continuous Newton s method: ẋ = B 1 Fx), t > x) = a 3) However, in order to find the solution, both the iterative as well as the continuous Newton s methods require the the inversion of the Jacobian matrix. On the one hand, inverting the Jacobian matrix in each iteration is computationally very expensive. On the other hand, for complex problems where the Jacobian matrix may

3 Solution of Post-Buckling & Limit Load Problems 545 Limit Points Load Displacement Figure 1: Newton s Method Fails at Limit Points be singular, such as near the limit-load points for geometrically nonlinear frames or in elastic-plastic solids, the iterative as well as the continuous Newtons s methods become problematic, as shown in Fig. 1. Various variants of the arc-length methods have been widely used for marching through the limit-points, in postbuckling analyses, such as those presented by [Wempner 1971); Riks 1972); Crisfield 1983); Lam and Morley 1992)]. These methods generally involve complex procedures by appending various constraints, and monitoring the eigenvalues of the Jacobian matrices. It will be advantageous to have a method to find the solutions for post-buckling problems of structures without inverting the Jacobian matrix, without using the arc-length method, and without worrying about initial guesses for the Newton s methods. In order to eliminate the need for inverting the Jacobian matrix in the Newton s iteration procedure, [Liu and Atluri 28a)] proposed an alternate first-order system of nonlinear ODEs, for the solution of the NAEs, Fx) =, by postulating an evolutionary equation for x, thus: ẋ = ν Fx), t > 4) qt) where ν is a nonzero constant and qt) may in general be a monotonically increasing function of t. In their approach, the term ν/qt) plays the major role of being

4 546 Copyright 214 Tech Science Press CMES, vol.98, no.6, pp , 214 a stabilizing controller to help obtain the solution even for a bad initial guess, and speeds up the convergence. This Fictitious Time Integration Method was successfully applied to the solution of various engineering problems in [Liu and Atluri 28a,b,c)]. In spite of its success, the Fictitious Time Integration Method was postulated only based on an engineering intuition, does not involve the Jacobian matrix at all, and was shown to only have local convergence. The homotopy method, as firstly introduced by [Davidenko 1953)], represents one of the best methods to enhance the convergence from a local convergence to a global convergence. Previously, all the homotopy methods were based on the construction of a vector homotopy function Hx, t), which serves the objective of continuously transforming a function Gx) into Fx) by introducing a homotopy parameter t t 1). The homotopy parameter t can be treated as a time-like fictitious variable, and the homotopy function can be any continuous function such that Hx,) = Gx) = and Hx,1) = Fx) =. Two of the most popular vector homotopy functions are the Fixed-point Homotopy Function: H F x,t) = tfx) + 1 t)x x ) =, t 1 5) and the Newton Homotopy Function: H N x,t) = tfx) + 1 t)[fx) Fx )] =, t 1 6) By using the vector homotopy method, the solution of the NAEs can be obtained by numerically integrating: ẋ = ) H 1 H x t, t 1 7) As can be seen in Eq. 7, the implementation of the Vector Homotopy Method necessitates the inversion of the matrix H x at each iteration. In order to remedy the shortcoming of the Vector Homotopy Method, [Liu, Yeih, Kuo, and Atluri 29)] proposed to solve the system of NAEs by constructing a Scalar Homotopy Function hx,t), such that hx,) = Gx) = and hx,1) = Fx) =. As an example, the following Scalar Fixed-point Homotopy Function was introduced in [Liu, Yeih, Kuo, and Atluri 29)]: hx,t) = 1 2 [t Fx) 2 1 t) x x 2 ], t 1 8) However, it may be more convenient to to define homotopy functions with t [, ] instead of t [,1], and use a positive and monotonically increasing function

5 Solution of Post-Buckling & Limit Load Problems 547 Qt) to enhance the convergence speed. In this paper, we consider the following two scalar homotopy functions, as denoted by the Scalar Fixed-point Homotopy Function: h f x,t) = 1 2 Fx) Qt) x x 2, t 9) and the Scalar Newton Homotopy function: h n x,t) = 1 2 Fx) Qt) Fx ) 2, t 1) respectively. By selecting a driving vector u so that the evolution of ẋ is parallel to u, the system of NAEs can be solved by numerically integrating: ẋ = h t h x ) u u, t 11) With different scalar homotopy functions hx,t), different Qt), and different driving vectors, u, Eq. 11 leads to different variants of scalar homotopy methods, see [Liu, Yeih, Kuo, and Atluri 29); Ku, Yieh, and Liu 21); Dai, Yue, and Atluri 214)]. In this paper, we select u such that u = h x. Thus, if h f is to be used, Eq. 11 leads to: ẋ = 1 Q F 2 QB T ) 2 QB T F + x x 2 F + x x, t 12) and if h n is to be used, we have: ẋ = 1 Q F 2 2 Q B T F 2 BT F, t 13) In this manuscript, Qt) = e t is used for simplicity, while various possible choices can be found in [Dai, Yue, and Atluri 214)]. For both of these two methods, the inversion of the Jacobian matrix is not involved. However, if a scalar equation is to be considered, Eq. 13 will fail at where B =. Therefore, Eq. 13 should only be used for non-scalar equations, while Eq. 12 can be used for general scalar and vector NAEs.

6 548 Copyright 214 Tech Science Press CMES, vol.98, no.6, pp , Solving the Williams Toggle with a Scalar Homotopy Method 2.1 The Classical Williams Solution The classical toggle problem, as introduced by [Williams 1964)], comprising of two rigidly jointed equal members of length l and angle β with respect to the horizontal axis, and subjected to an externally applied vertical load W at the apex, is shown in Fig. 2. The structure deforms in a symmetrical mode as shown in Fig. 3 with the deflected position of the neutral axis of member rs denoted by r s, r β l W s l Figure 2: The classical Williams toggle r,r s W s δ Figure 3: Symmetrical deformation of toggle Following the same assumptions and nomenclature in [Williams 1964)], the externally applied load W can be expressed in terms of the deformation at the apex, δ, through the following series of equations, W 2 F + Psinβ 14) where, F is the component of the reaction force perpendicular to the undeflected position of the neutral axis denoted here by rs and P is the component of the force at the end of the member parallel to rs. P is expressed in terms of δ as: P = AE l δ sinβ.6 δ 2 ) l 15)

7 Solution of Post-Buckling & Limit Load Problems 549 where, AE is the extensional rigidity of the member. F is then expressed in terms of δ using nonlinear elastic stability theory as: F = 6EI l 2 d δ 5 l where d 5 can be obtained by the following relations: 16) d 5 = 2d 4 wρ) d 4 = d 3 3 d 3 = d 1 + d 2 d 2 = d 1 wρ) d 1 = 1 [ π 2 ρ 2 41 wρ)) + wρ) wρ) = π 2 ρ = π2 EI l 2 ρ cot π 2 ρ ] Combining Eq. 14 through Eq. 17, for a given load W, the vertical displacement, δ, of the Williams toggle can be found by solving the following scalar nonlinear algebraic equation: 3AEsin β 5l 2 δ 2 + 6EId 5 l 3 + AEsinβ 2 l ) 17) δ W 2 = 18) In this study, the set of parameters l,ei,ae are considered to be the same as those presented in [Williams 1964)], and are given in Tab. 1. By changing the height of the apex of the toggle, three cases of interest are generated, as shown in Fig. 4. The first case takes l sinβ =.32, which represents the original plot in [Williams 1964)]. The second and third cases, l sin β =.38 and l sin β =.44, respectively, show the effect of raising the apex on the load-deflection curve of the toggle, and introduce limit points in the scalar NAE at which the Jacobian is singular. 2.2 Solving Williams Equation with a Scalar Homotopy Method In order to characterize the deflection δ resulting from a specific external load, the scalar NAE Eq. 18) must be solved. To better understand the limitations on solving the NAE utilizing the classical Newton s method, the behavior of the Jacobian derived analytically from Eq. 18 is shown in Fig. 5 for the three values of l sinβ introduced earlier.

8 55 Copyright 214 Tech Science Press CMES, vol.98, no.6, pp , 214 Table 1: Parameters set in [Williams 1964)] Parameter Value Units l in EI lb/in 2 EA lb W lb) δ in) l sinβ =.32 l sinβ =.38 l sinβ =.44 Figure 4: Three Cases of Load-Deflection Curves for Williams Toggle Limit-points are those where the Jacobian becomes close to zero and thus classical Newton s method will fail. For this end, the previously introduced Scalar Homotopy Methods are used to avoid inverting the Jacobian for the solution of the NAE. As discussed in Section 1, the Scalar Newton Homotopy Method Eq. 13) only works for a system of NAEs. Thus, the Scalar Fixed-point Homotopy Method Eq. 12) is adopted here for the solution of the scalar NAE of the Williams toggle. Setting the tolerance to 1 6 for the original Williams toggle with l sinβ =.32, the Williams equation can be solved for an arbitrary load, here chosen as W = lb. A comparison between Newton s Method and the Scalar Fixed-point Homotopy Method is shown in Tab. 2 The comparison shows the fast convergence speed of Newton s method achieving the required accuracy in just 9 iterations whereas it took the Scalar Fixed-point

9 Solution of Post-Buckling & Limit Load Problems The Jacobian, B l sinβ =.32 l sinβ =.38 l sinβ = δ in) Figure 5: Scalar Jacobian Evaluation of Williams Scalar NAE Table 2: Solution of Original Williams Equation with No Limit Points Method No. Iteration, N Achieved Accuracy Newton s Method N = Scalar Fixed-point Homotopy Method N = Homotopy Method 3 iterations to achieve similar accuracy results. This case of l sin β =.32 as shown in Fig. 5 has no singularities in the Jacobian, thus the superior performance of Newton s method is expected. Fig. 6 show the evolution of the solution and the fast convergence of Newton s Method as compared to the Scalar Fixed-point Homotopy Method. A second case is considered for l sinβ =.44 where the Williams equation is solved for an externally applied load selected near the limit point, W = lb. Both Newton s Method and the Scalar Fixed-point Homotopy Method are utilized, and the results are shown in Tab. 3 After 1 iterations Newton s method did not converge to the solution whereas the Scalar Fixed-point Homotopy Method achieved the required accuracy in 345 iterations. Fig. 7 shows a comparison between the evolution of the solution for the two methods. It is shown that Newton s method will keep fluctuating about

10 552 Copyright 214 Tech Science Press CMES, vol.98, no.6, pp , 214 the solution and not converge to achieve the required accuracy, whereas the Scalar Fixed-point Homotopy Method converges to the solution with the required high accuracy Vertical Deflection, δ Vertical Deflection, δ No. Iterations, N No. Iterations, N a) Newton s Method b) Scalar Fixed-point Homotopy Method Figure 6: Vertical Deflection vs. No. Iterations, l sin β =.32 Table 3: Solution of Williams Equation for Loading near Limit Point Method No. Iteration, N Achieved Accuracy Newton s Method N = 1.44 Scalar Fixed-point Homotopy Method N = Vertical Deflection, δ Vertical Deflection, δ No. Iterations, N No. Iterations, N a) Newton s Method b) Scalar Fixed-point Homotopy Method Figure 7: Vertical Deflection vs. No. Iterations, l sin β =.44

11 Solution of Post-Buckling & Limit Load Problems Application to Finite Element Analyses 3.1 A Generalized Finite Element Model for Frame Structures The currently adapted Scalar Homotopy Methods can be easily combined with general purpose nonlinear finite element programs, by taking the directly derived tangent stiffness matrix at each iteration as the Jacobian matrix, and taking the difference between generalized internal force vector and the external force vector the residual) as Fx). In this manuscript, explicitly derived tangent stiffness matrices and nodal forces of large-deformation beam-column members are adopted following the work of [Kondoh and Atluri 1986)]. The basic derivations of [Kondoh and Atluri 1986)] are briefly reviewed here. First, the nomenclature and the sign convention used in the derivation for a general beam column member are shown in Fig. 8. The functions wz) and uz) describe the displacement at the centroidal axis of the element along the z and the x axes, respectively. The angles θ1 and θ 2 are the angles between the tangent to the deformed centroidal axis and the line joining the two nodes of the deformed element at nodes 1 and 2, respectively. M 1 and M 2 are the bending moments at the two nodes and N is the axial force in the beam member. The total rotation of the beam member is then given by, θ = θ + θ 19) where, θ describes the rigid rotation of the beam member and is measured between the line joining the two nodes of the deformed beam and the z-axis. θ is expressed in terms of the nodal displacements as, ) ũ θ = tan 1 2) l + w where, ũ = u 2 u 1 and w = w 2 w 1. From Eq. 19 and Eq. 2, the non-rigid rotations at the two nodes, θ1 and θ 2, are given by, ) ũ θ1 = θ 1 tan 1 l + w ) 21) ũ θ2 = θ 2 tan 1 l + w The total stretch/deformation of the beam member is then expressed in terms of the displacements at the two nodes as, [ δ = l + w) 2 + ũ 2] 1/2 l 22)

12 554 Copyright 214 Tech Science Press CMES, vol.98, no.6, pp , 214 xu) Beam Member Before Deformation 1 zw) 2 l Beam Member after Deformation & Sign Convention N M 1 θ1 M 2 1 S 2 θ 2 N Q Q δ Figure 8: Kinematics & Nomenclature for a Beam Member l The axial force and bending moment are non-dimensionalized through, n = Nl2 EI, m = MI EI 23) The non-rigid rotation and the non-dimensional bending moment are decomposed into symmetric and anti-symmetric parts given by, θ a = 1 2 θ 1 + θ 2 ), θ s = 1 2 θ 1 θ 2 ) m a = m 1 m 2 ), m s = m 1 + m 2 ) 24) The relation between the generalized displacements and forces at the nodes of the beam member is given by, θ a = h a m a, δ l = 1 2 dha dn θs = h s m s ) θ 2 a h a 2 dhs dn ) θ 2 s h 2 s + N EA 25)

13 Solution of Post-Buckling & Limit Load Problems 555 where, h a and h s are given by, for n ) ) h a = 1 n 1 n) 1/2 1 n) 1/2 cot, h 2 n) 1/2 s = tan 2 2 n) 1/2 2 for n > h a = 1 n 1 coth 2n) 1/2 n) 1/2 2 ), h s = 1 tanh 2n) 1/2 n) 1/2 2 ) 26) The kinematics variables can then be expressed in a vector form for a beam member m as, {d m } = [ w 1 w 2 u 1 u 2 θ 1 θ 2 ] T 27) The increment of the internal energy of a beam member is then expressed in terms of the increment of kinematics variables vector, {d m }, the tangent stiffness matrix [K m ] and the internal force vector {R m } as, π = 1 2 { dm } T [K m ]{ d m } + { d m } T {R m } 28) The tangent stiffness matrix, [K m ], and the internal force vector, {R m }, for the member m are given by, [K m ] = [A dd ] 1 {A nd }{A nd } T A nn 29) {R m } = {B d } 1 {A nd } A nn 3) where the elements constructing Eq. 29 and Eq. 3 are given by, [A dd ] = N 2 δ + EI l + M 2 θ w 2 a a w ) ) 2 2 [E] 1 θ a h a w Symmetric N 2 δ w ũ + M a 2 θa w ũ )) EI [E] + EI 1 θ a l h a w N 2 δ + EI l θ a ũ + M 2 θ w 2 a a w 2 1 θ a h a w θa 2lh a w {I} ) 2 ) [E] EI 2lh a θ a ũ {I} EI 1 4l + 1 ha hs ) EI 4l EI θa 2lh a w {I} EI θa 2lh a ũ {I} ) 1 1 ha hs ) ha hs EI 4l 31)

14 556 Copyright 214 Tech Science Press CMES, vol.98, no.6, pp , 214 {A nd } = ) δ w + d 1 θ a dn h a w θ a ) δ ũ + d 1 θ a dn h a ũ θ a ) 1 d 1 2 dn h a θa + d 1 ) dn h s 1 d 1 2 dn h a θa d 1 dn h s ) {I} ) {I} ) ) ) θ s θ s ) ) A nn = l3 d 2 1 2EI dn 2 θa 2 + d2 1 h a dn 2 h s ) N δ w + M a θ a w {I} ) {B d } = N δ ũ + M a θ a ũ {I} 1 2 M a + M s ) 1 2 M s M a ) B n = δ + 1 ) d 1 θa 2 l + 1 ) d 1 2 dn h a 2 dn h s { } [ ] {I} = [E] = ) ) θs 2 l EA θs 2 l Nl EA The load-deflection curve generated using the finite element model in Eq. 28 is compared against the original Williams problem with l sinβ =.32 in Fig. 9. Other cases with l sinβ =.38 and l sinβ =.44 are shown in Fig. 1 and Fig. 11, respectively. The Scalar Fixed Point Homotopy Method, Eq. 9, is used to generate the load-deflection curves for the finite element model for all three cases. As shown, the finite element model accurately describe the load-deflection characteristics of the Williams toggle as it agrees well with the solutions of the scalar NAE presented in [Williams 1964)] and summarized in Eqs The Scalar Fixedpoint Homotopy Method method successfully solved the FEM equations capturing the load-deflection relation around the limit points, at which the Newton s method fails to find the solution, as will be shown in this next subsection. 3.2 Solution of the Finite Element Model Using Scalar Homotopy Methods The Scalar Fixed-point Homotopy Method, Eq. 12, and the Scalar Newton Homotopy Method, Eq. 13, are both applied to the finite element model to solve for the deflection given a specific load. As done in the previous section the case of l sinβ =.44 is examined with the same value of the load applied near the limit point. Setting the tolerance for the relative residual error to be 1 6, the two methods are compared with Newton s method and the results are shown in Tab. 4. Both Scalar Homotopy Methods proved superior to the Newton s method, as both converged to the solution with the required accuracy whereas the Newton s method 32) 33) 34) 35) 36)

15 Solution of Post-Buckling & Limit Load Problems Williams Equation Finite Element Model 6 5 W lb) δ in) Figure 9: Load-Deflection, Williams Equation & Finite Element, l sinβ = Williams Equation Finite Element Model 4 W lb) δ in) Figure 1: Load-Deflection, Williams Equation & Finite Element, l sinβ =.38

16 558 Copyright 214 Tech Science Press CMES, vol.98, no.6, pp , Williams Equation Finite Element Model 5 W lb) δ in) Figure 11: Load-Deflection, Williams Equation & Finite Element, l sinβ =.44 Table 4: Solution of Finite Element Model for Loading near Limit Point Method No. Iteration, N Achieved Accuracy Newton s Method N = Scalar Fixed-point Homotopy Method N = Scalar Newton Homotopy Method N = failed to find the solution after 1 iterations. A zoomed in plot is shown in Fig. 12 to illustrate the oscillating behavior of Newton s method and its failure to find the solution. The Scalar Fixed-point Homotopy Method converged in 16 iterations Fig. 13), which is about one third the number of iterations required by the Scalar Newton Homotopy Method, Fig. 14). This makes the Scalar Fixedpoint Homotopy Method more suitable for solving the problem of Williams toggle, whereas the Scalar Newton Homotopy Method provides a valid alternative to obtain the solution. The Scalar Homotopy Methods developed in this work and in previous works are suitable to solve general nonlinear finite element models with very high accuracy, without inverting the tangent stiffness matrix, and without having to use the computationally expensive arc-length methods. In order to illustrate the efficiency of the scalar homotopy methods the tolerance

17 Solution of Post-Buckling & Limit Load Problems Residual Error No. I terations, N Figure 12: Residual Error in Newton s Method Residual Error No. I terations, N Figure 13: Residual Error in Scalar Fixed-point Homotopy Method

18 56 Copyright 214 Tech Science Press CMES, vol.98, no.6, pp , Residual Error No. I terations, N Figure 14: Residual Error in Scalar Newton Homotopy Method Table 5: Solution of Finite Element Model for Loading near Limit Point Method No. Iteration, N Achieved Accuracy Newton s Method N = Fixed-point Homotopy Method N = Newton Homotopy Method N = for the relative residual error is relaxed to match existing finite element solvers.1%). For this case an external load of 44 lb. is applied and the results are shown in Tab. 5. The two methods achieved the required accuracy within 14 iterations, which demonstrates the power of the scalar homotopy methods in solving engineering problems and the fast convergence that can be achieved when addressing such problems. The Newton s method failed after 1 iterations, with the same oscillatory non-convergent behavior shown in Fig. 12. Figs show the path to convergence of the Scalar Fixed-point Homotopy Method and the Scalar Newton Homotopy Method, respectively.

19 Solution of Post-Buckling & Limit Load Problems Residual Error No. I terations, N Figure 15: Residual Error in Scalar Fixed-point Homotopy Method Residual Error No. I terations, N Figure 16: Residual Error in Scalar Newton Homotopy Method

20 562 Copyright 214 Tech Science Press CMES, vol.98, no.6, pp , Conclusion The Scalar Homotopy Method is applied to the solution of post-buckling and limit load problems of plane frames considering geometrical nonlinearities. Explicitly derived tangent stiffness matrices and nodal forces of large-deformation beamcolumn members are adopted following the work of [Kondoh and Atluri 1986)]. By using the Scalar Homotopy Method, nodal displacements of the equilibrium s- tate are iteratively solved for, without inverting the Jacobian tangent stiffness) matrix and without using complex arc-length methods. This simple method thus saves computational time and avoids the problematic behavior of the Newton s method when the Jacobian matrix is singular. While the simple Williams toggle is considered in this paper, extension to general finite element analyses of space frames, plates, shells and elastic-plastic solids will be considered in forthcoming studies. Acknowledgement: This work is supported by the Texas A&M Institute for Advanced Study TIAS). It was initiated while S.N. Atluri visited TIAS briefly in January, 214. The support of various U.S. government agencies during is also thankfully acknowledged. Messrs Le and Riddick of ARL are thanked for their encouragement. References Crisfield, M. A. 1983): An arc-length method including line searches and accelerations. International journal for numerical methods in engineering, vol. 19, no. 9, pp Dai, H.; Yue, X.; Atluri, S. N. 214): Solutions of the von kármán plate equations by a galerkin method, without inverting the tangent s- tiffness matrix. Journal of Mechanics of Materials and Structures, vol. 9. dx.doi.org/1.214/jomms Davidenko, D. F. 1953): On a new method of numerical solution of systems of nonliear equations. Dokl. Akad. Nauk SSSR, vol. 88, pp Hirsch, M. W.; Smale, S. 1979): On algorithms for solving f x)=. Communications on Pure and Applied Mathematics, vol. 32, no. 3, pp Kondoh, K.; Atluri, S. N. 1986): A simplified finite element method for large deformation, post-buckling analyses of large frame structures, using explicitly derived tangent stiffness matrices. International journal for numerical methods in engineering, vol. 23, no. 1, pp

21 Solution of Post-Buckling & Limit Load Problems 563 Ku, C.-Y.; Yieh, W.; Liu, C.-S. 21): Solving non-linear algebraic equations by a scalar newton-homotopy continuation method. International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 6, pp Lam, W.; Morley, C. 1992): Arc-length method for passing limit points in structural calculation. Journal of structural engineering, vol. 118, no. 1, pp Liu, C.-S.; Atluri, S. N. 28a): A novel time integration method for solving a large system of non-linear algebraic equations. CMES: Computer Modeling in Engineering & Sciences, vol. 31, no. 2, pp Liu, C.-S.; Atluri, S. N. 28b): A novel fictitious time integration method for solving the discretized inverse sturm-liouville problems, for specified eigenvalues. CMES: Computer Modeling in Engineering & Sciences, vol. 36, no. 3, pp Liu, C.-S.; Atluri, S. N. 28c): A fictitious time integration method ftim) for solving mixed complementarity problems with applications to non-linear optimization. CMES: Computer Modeling in Engineering & Sciences, vol. 34, no. 2, pp Liu, C. S.; Yeih, W.; Kuo, C. L.; Atluri, S. N. 29): A scalar homotopy method for solving an over/under determined system of non-linear algebraic equations. CMES: Computer Modeling in Engineering and Sciences, vol. 53, no. 1, pp Riks, E. 1972): The application of newton s method to the problem of elastic stability. Journal of Applied Mechanics, vol. 39, no. 4, pp Wempner, G. A. 1971): Discrete approximations related to nonlinear theories of solids. International Journal of Solids and Structures, vol. 7, no. 11, pp Williams, F. W. 1964): An approach to the non-linear behaviour of the members of a rigid jointed plane framework with finite deflections. The Quarterly Journal of Mechanics and Applied Mathematics, vol. 17, no. 4, pp

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